Journal of the Physical Society of Japan Vol 72, No 5, May, 2003, pp 1254–1259 #2003 The Physical Society of Japan A New Anharmonic Factor and EXAFS Including Anharmonic Contributions Nguyen Van H UNGÃ, Nguyen Ba D UCy and Ronald R FRAHMz Bergische Universitaet-Gesamthochschule Wuppertal, FB: 8-Physik, Gauss-Strasse 20, 42097 Wuppertal, Germany (Received October 25, 2002) A new anharmonic factor and the extended X-ray absorption fine structure (EXAFS) including anharmonic contributions have been developed based on the cumulant expansion and the single-shell model Analytical expressions for the anharmonic contributions to the amplitude and to the phase of the EXAFS have been derived The EXAFS and its parameters contain anharmonic effects at high temperature and approach those of the harmonic model at low temperature Numerical results for Cu agree well with experiment Peaks in the Fourier transform of the calculated anharmonic EXAFS for the first shell at 297 K and 703 K agree well with the experimental ones and are shifted significantly compared to those of the harmonic model KEYWORDS: anharmonic EXAFS, cumulants, temperature dependence DOI: 10.1143/JPSJ.72.1254 Introduction The harmonic approximation in EXAFS calculations works very well1) at low temperatures because the anharmonic contributions to atomic thermal vibrations can be neglected But at different high temperatures the EXAFS spectra provide apparently different structural information2–17) due to the anharmonicity, and these effects need to be evaluated Moreover, for some aspects like catalysis research the EXAFS studies carried out at low temperature may not provide a correct structural picture and the hightemperature EXAFS, where the anharmonicity must be included, is necessary.2) The formalism for including anharmonic effects in EXAFS is often based on the cumulant expansion approach,3,5) according to which the EXAFS oscillation function is described by ( " #) X ð2ikÞn e2R=kị iẩkị nị Im e exp 2ikR ỵ kị ẳ FðkÞ kR2 n! n ð1Þ where FðkÞ is the real atomic backscattering amplitude, È is the net phase shift, k and are the wave number and the mean free path of the photoelectron, respectively, and ðnÞ (n ¼ 1; 2; 3; ) are the cumulants They appear due to the thermal average of the function exp ði2krÞ in which the asymmetric terms are expanded in a Taylor series about R ¼ hri with r as the instantaneous bond length between absorbing and backscattering atoms and then are rewritten in terms of cumulants Based on this approach the anharmonic effects in EXAFS have been often valuated by the ratio methods.3–9) Another way is the direct calculation and analysis of EXAFS and its parameters including anharmonic effects at any temperature For this purpose an anharmonic factor has been introduced10–12) to take into account the anharmonic contributions to the mean square relative displacement (MSRD) This procedure provides a good agreement with experiment,11) but the expressions for the anharmonic factor and à E-mail: vhung@phys-hu.edu.vn Permanent address: Department of Physics, Hanoi National University, 334 NguyenTrai, Hanoi, Vietnam E-mail: nbduc@netnam.vn z E-mail: frahm@uni-wupprtal.de for the phase change of the EXAFS due to anharmonicity contain a fitting parameter, and the cumulants were obtained by an extrapolation procedure from the experimental data This work firstly is a next step of ref 11 to develop an analytical procedure which overcomes the above mentioned limitations and to show more information Our further development is the derivation of analytical expressions for the anharmonic factor determining the anharmonic contributions to the amplitude and for the anharmonic contributions to the phase of the EXAFS The cumulants contained in the derived expressions can be considered by several procedures.4,6,7,13–16) In this work the quantum statistical approach with anharmonic correlated Einstein model15) has been used for calculation of the cumulants in which the parameters of the anharmonic effective potential are based on a Morse potential that characterizes the interaction between each pair of atoms Including contributions of all atoms in all directions in a small cluster by a simple way15) this model avoids full lattices dynamical14) or dynamical matrix16) calculations jet provides reasonable agreement with experiment and with the other theory results14) even for the case of strongly anharmonic crystal Cu This model also is successful in extracting physical parameters from the EXAFS measured data,27) as well as in the investigation of local force constants of transition metal dopants in a Nickel host,28) and in contribution to theoretical approaches to the EXAFS.29) Moreover, for nanostructure the clusters become too small the bulk theory may start to break down, which is one place the small cluster approach15) input is necessary.29) This work secondly is a next step of the work by Hung and Rehr15) applying the anharmonic correlated Einstein model to calculation and analysis of EXAFS and its parameters including anharmonic contributions We get the total MSRD and the EXAFS function which include anharmonic effects at high temperatures and are approaching those of the harmonic model at low temperatures Cu metal spectra, which are often used for testing new theories2,11,14,15,18,27) also have been considered in this work, and numerical results are found to be in good agreement with experiment.17,19) y 1254 J Phys Soc Jpn., Vol 72, No 5, May, 2003 N V HUNG et al Formalism The EXAFS oscillation function eq (1) including anharmonic effects contains the Debye–Waller factor eÀWðk;TÞ accounting for the effects of the thermal vibrations of atoms Based on the analysis4,15) of cumulant expansion we obtain 1 ð1Þ ð2Þ ð2Þ À Wðk; TÞ ¼ 2ik ðTÞ À 2k ðTÞ À 4ik ðTÞ R ðkÞ À ð3Þ ik ðTÞ þ ð4Þ ðTÞk4 þ Á Á Á ; 3 ð2Þ where ð1Þ is the first cumulant or net thermal expansion, ð2Þ is the second cumulant which is equal to the MSRD , ð3Þ and ð4Þ are the third and the fourth cumulants, respectively, the remaining parameters were defined above The higher cumulants are not included due to their small contributions.3,5) To consider anharmonic contributions to the MSRD we used an argument analogous to the one20) for its change due to the temperature increase and obtain ðTÞ À ðT0 ị ẳ ỵ TịịẵH2 T0 ị; V Tị ẳ 2
G V 3ị where G is Gruăneisen parameter, and ÁV=V is the relative volume change due to thermal expansion, T0 is a very low temperature so that ðT0 Þ is a harmonic MSRD This result agrees with the one in another consideration4) on the change of the MSRD Developing further eq (3) we obtain the total MSRD Tị ẳ H2 Tị ỵ TịẵH2 À ðT0 Þ; ð4Þ It is clear that the MSRD approaches the very small value of zero-point contribution 02 when the temperature approaches zero, i.e., ðT0 Þ ! 02 ; for T0 ! 0: Hence, it can be seen in eq (4) that the total MSRD ðTÞ at a given temperature T consists of the harmonic contribution H2 ðTÞ and the anharmonic one A2 Tị Tị ẳ H2 Tị ỵ A2 Tị; A2 Tị ẳ TịẵH2 Tị 02 ; 5ị This separation will help us to determine the anharmonic contribution to the EXAFS amplitude We will illustrate the theory for a simple fcc crystal, though the generalization to other structures or longer-range interactions is straightforward In the present approach we apply the anharmonic correlated Einstein model15) to the calculation of cumulants where the effective potential is given by Veff ðxÞ $ ẳ keff x2 ỵ k3 x3 ỵ Á X ^ 12 ; R ^ ij ; ẳ M1 M2 : ẳ Vxị ỵ V xR Mi M1 ỵ M2 j6ẳi 6ị Here x is the deviation of instantaneous bond length between ^ is the bond unit vector, keff is two atoms from equilibrium, R 1255 effective spring constant, and k3 the cubic parameter giving an asymmetry in the pair distribution function The correlated Einstein model may be defined as a oscillation of a pair of atoms with masses M1 and M2 (e.g., absorber and backscatterer) in a given system The contributions of all their neighbors in all directions in a small cluster are given by the last term in the left-hand side of eq (6), where the sum i is over absorber (i ¼ 1) and backscatterer (i ¼ 2), and the sum j is over all their near neighbors, excluding the absorber and backscatterer themselves whose contributions are described by the term VðxÞ To model the asymmetry we replaced the harmonic potential by an anharmonic one, e.g., a Morse potential21) with parameters D and which charactrizes the interaction of each pair of atoms Applying it to the effective potential of the system of eq (6) (ignoring the overall constant) we obtain keff ¼ 5D À a ẳ !2E ; 7ị h" !E k3 ¼ À D 3 ; E ¼ ; kB where kB is the Boltzmann constant; !E ; E are the correlated Einstein frequency and temperature Using the above results in first-order thermodynamic perturbation theory13,15) with consideration of the phonon– phonon interaction for taking into account the anharmonicity we obtain the cumulants 1ỵz 3 2ị ; 01ị ẳ 1ị Tị ẳ aTị ẳ 01ị 8ị 1z 1ỵz h" !E ; 02 ẳ Tị ẳ 02 ; z ẳ eE =T 9ị 1z 10D 2 3ị Tị ẳ 03ị Tị ỵ 10z ỵ z2 ; zị2 03ị ẳ 22 ị ; ð10Þ where 0ð1Þ ; 02 ; 0ð3Þ are the zero-point contributions to the first, second and third cumulant, respectively We calculated the relative thermal volume change ÁV=V using RTị ẳ R ỵ aTị and Gruăneisen parameter G ẳ À @@lnln!VE By substituting the obtained results in eq (3) we derived an anharmonic factor ! 9ðTÞkB T 3kB T 3kB T 1ỵ 1ỵ ; Tị ẳ 16D 8DR 8DR 11ị 2eE =T Tị ẳ : ỵ eÀE =T This factor is proportional to the temperature and inversely proportional to the shell radius, thus reflecting a similar property of anharmonicity obtained in an experimental catalysis research2) if R is considered as particle radius The anharmonic contribution ÈA to the EXAFS phase at a given temperature is the difference between the total phase and the one of the harmonic EXAFS On the left-hand side of eq (2) the 2nd and the 5th terms contribute to the EXAFS amplitude Only the 1st, the 4th terms and the anharmonic contributions to the MSRD in the 3rd term are the anharmonic contributions to the phase Therefore, from this equation we obtain 1256 J Phys Soc Jpn., Vol 72, No 5, May, 2003 1ị A T; kị ẳ 2k Tị 2A2 ðTÞ N V HUNG et al ! 1 ð3Þ À À ðTÞk : R kị 12ị k; Tị ẳ X S2 Nj j kR2j Fj ðkÞeÀð2k The 4th cumulant is often very small.13,15,17) This is why we obtained from eqs (1) and (2), taking into account the above results, the temperature dependent K-edge EXAFS function including anharmonic effects as 2 sin2kRj ỵ j kị ỵ jA k; Tịị Tịỵ2Rj =ðkÞÞ ð13Þ which by taking eq (5) into account is resulting in X S2 Nj À Á 2 Fj kịe2k ẵH TịỵA Tịỵ2Rj =kịị sin 2kRj ỵ j kị ỵ jA k; Tị k; Tị ẳ j kRj # À1 ) (A keff (N/m) !E (1013 Hz) E (K) Cu Al 0.3429 0.2703 1.3588 1.1646 50.7478 29.3686 3.0889 3.6102 235.9494 275.7695 Ni 0.4205 1.4149 67.9150 3.7217 284.3095 Anharmonic factor β(T) 0.12 Cu Al Ni 0.10 0.08 0.06 0.04 0.02 0.00 100 200 300 400 500 600 700 T(K) Fig Temperature dependence of the calculated anharmonic factor ðTÞ for Cu, Al and Ni 0.0018 0.0016 Cu 0.0014 0.0012 (Å ) We applied the expressions derived in the previous section to numerical calculations for Cu, Al, and Ni Their Morse potential parameters23) D, ; calculated effective spring constant keff , correlated Einstein frequency !E and temperature E are written in Table I, where our calculated value E % 236 K for Cu agrees well with the measured one of 232(5) K.9) Figure shows the temperature dependence of our calculated anharmonic factors ðTÞ for Cu, Al and Ni In the case of Cu it has the values 0.028 at 300 K and 0.084 at 700 K which agree well with those obtained in the other studies.11,25) Figure shows the temperature dependence of our calculated anharmonic contribution A2 ðTÞ to the MSRD determining the anharmonic contribution to the EXAFS amplitude of Cu It is small at low temperatures and then increases strongly at high temperatures having a form looking like the one of the third cumulant (Fig 5) This result also shows that below 100 K no anharmonic effect in the EXAFS of Cu is expected It agrees well with our D (eV) A Discussion of Numerical Results and Comparison with Experiment Crystal Table I Morse potential parameters D, , the calculated effective spring constant keff , correlated Einstein frequency !E , Einstein temperature E of Cu, Al and Ni σ where S20 is the square of the many body overlap term, Nj is the atomic number of each shell, the remaining parameters were defined above, the mean free path is defined by the imaginary part of the complex photoelectron momentum p ẳ k ỵ i=, and the sum is over all atomic shells It is obvious that in eq (14) A2 ðTÞ determines the anharmonic contribution to the amplitude characterizing the attenuation, and ÈA ðk; TÞ is the anharmonic contribution to the phase characterizing the phase shift of EXAFS spectra They are calculated by eqs (5) and (12), respectively Their values characterize the temperature dependence of the anharmonicity, but the anharmonicity is described by the cumulants given by eqs (8)–(10) obtained by consideration of the phonon–phonon interaction process That is why they also characterize the temperature dependence of the phonon–phonon interaction in the EXAFS theory At low temperatures these anharmonic values approach zero and the EXAFS function eq (14) is reduced to the one of the harmonic model The Morse potential parameters D and can be obtained using experimental values of the energy of sublimation, the compressibility, and the lattice constant, which are known already.22) A such method for calculation of the Morse potential parameters has been developed for cubic crystals23) and for other structures.24) ð14Þ 0.0010 0.0008 0.0006 0.0004 0.0002 0.0000 100 200 300 400 500 600 700 T(K) Fig Temperature dependence of the calculated anharmonic contribution A2 ðTÞ to the MSRD determing the anharmonic contributions to the EXAFS amplitude J Phys Soc Jpn., Vol 72, No 5, May, 2003 N V HUNG et al 0.022 0.0008 0.020 Cu 0.018 0.0006 σ(3)(Å3) 0.014 2 Cu total (anharmonic) harmonic exp (Refs 17, 19) exp (Ref 19) 0.016 σ (Å ) 1257 0.012 0.010 Present theory exp (Refs 17, 19) 0.0004 0.008 0.0002 0.006 0.004 0.002 0.0000 100 200 300 400 500 600 700 100 200 300 400 500 600 700 T(K) T(K) Fig Temperature dependence of the calculated total MSRD ðTÞ of Cu compared to the harmonic one H2 ðTÞ and to the measured values at 295 K17,19) and at 700 K.19) Fig Temperature dependence of the calculated third cumulant ð3Þ ðTÞ of Cu compared to the measured value at 295 K.17,19) 0.022 0.020 Cu 0.018 Present theory σ(1)(Å) 0.016 0.014 exp (Ref 19) 0.012 0.010 0.008 0.006 0.004 0.002 100 200 300 400 500 600 700 T(K) previous prediction11,12) and with experiment.2,17) Therefore, A2 also makes it possible to determine the temperature above which the anharmonic effects or the phonon–phonon interaction are visible For Cu this temperature is about 100 K The increase of the anharmonic contribution to the EXAFS amplitude at high temperature characterizes the attenuation of EXAFS spectra in comparison to the one calculated by the harmonic model (Fig 7) Figure illustrates the temperature dependence of our calculated total MSRD ðTÞ of Cu compared to its harmonic one H2 ðTÞ and to the experiment.17,19) The difference between the total and the harmonic values becomes visible at 100 K, but it is very small and can be important only from about room tempera# of the total ture Our calculated value 8:67 Â 10À3 A anharmonic MSRD at 295 K agree well with the measured # and with other theory14) result of one17,19) of 8:67 Â 10À3 A # Our result at 700 K also agrees well with 5:20 Â 10À3 A 19) experiment The temperature dependence of the calculated first cumulant ð1Þ and third cumulant ð3Þ of Cu are illustrated in Figs and 5, respectively They contribute to the phase shifts of the EXAFS due to anharmonicity (Fig 6) Theoretical results agree well with the experimental values at 295 K for ð1Þ 19) and for ð3Þ 17,19) Figure illustrates the temperature and k-dependence of our calculated anharmonic contribution ÈA ðk; TÞ to the EXAFS phase of Cu for the first Fig Temperature and k-dependence of the calculated anharmonic contribution ÈA ðT; kÞ to the EXAFS phase of Cu 20 Cu, 1st shell, single scattering 15 295K, harmonic 295K,anharmonic 700K, harmonic 700K, anharmonic 10 χk3 Fig Temperature dependence of the calculated first cumulant ð1Þ ðTÞ or net thermal expansion of Cu compared to the measured value at 295 K.19) -5 -10 -15 -20 10 15 20 -1 k(Å ) Fig Comparison of the calculated EXAFS spectrum of Cu at 295 K and 700 K including the anharmonic contribution to the one calculated by the harmonic model for the first shell for single scattering shell for single scattering These contributions are especially large at high temperatures and high k-values They contribute to the phase differences between the calculated anharmonic EXAFS spectra at different high temperatures and to their phase shifts compared to those calculated by the harmonic model Figure shows the difference between the 1258 J Phys Soc Jpn., Vol 72, No 5, May, 2003 EXAFS spectra k3 of Cu at 295 K and 700 K calculated by the harmonic FEFF code1) and those including anharmonic contributions The anharmonic spectra are shifted to the left and attenuated especially at high k-values Fourier transform # À1 < k < 13 A # À1 for magnitudes over the range 2:5 A T ¼ 297 K [Fig 8(a)] and for T ¼ 703 K [Fig 8(b)] of EXAFS spectra of Cu calculated by the present anharmonic theory are compared to those calculated by the harmonic FEFF code1) and to the experimental results,19) measured at HASYLAB (DESY, Germany) For XANES the multiple scattering is important, but for EXAFS the single scattering is dominant,26) and the main contribution to EXAFS is given by the first shell.7) This is why for testing the theory only the calculated EXAFS of the first shell for single scattering has been used for the comparison to the experiment The generalization to the other shells is straightforward Our calculated EXAFS Fourier transform magnitudes of Cu including anharmonic contributions for the first shell agree well with the measured ones They are shifted to smaller # at 297 K and by 0.07 A # at 703 K in distances by 0.03 A N V HUNG et al comparison to the harmonic model results, as well as yielding apparently different structural information at the different high temperatures Conclusion A new analytical procedure for calculation and analysis of EXAFS and its parameters including anharmonic contributions has been developed based on the cumulant expansion and the single-shell model Our development is the derivation of the expressions for the anharmonic contributions to the amplitude and to the phase of the EXAFS Total MSRD is the sum of the harmonic and the anharmonic ones The anharmonic contribution to the MSRD is obtained by multiplication of the harmonic MSRD with the new derived anharmonic factor which characterizes anharmonic contribution to the EXAFS amplitude Anharmonic contributions to the EXAFS and its parameters such as amplitude, phase, Fourier transform magnitude and the cumulants can be calculated and analyzed for any temperature and for any k-value The expressions derived for the EXAFS and its parameters include anharmonic contributions at high temperatures and are approaching those of the harmonic model at low temperature Moreover, based on the anharmonic contribution to the EXAFS amplitude we also can predict the temperature above which the anharmonicity or the phonon–phonon interaction in the EXAFS is visible Therefore, this work not only shows the advantages of the analytical procedure towards the ab initio calculation of the EXAFS and its parameters including anharmonic contributions, but also provides more useful and suggesting information compared to the previous empirical procedure Based on the anharmonic correlated Einstein model the calculating procedure is simplified yet provides a good agreement of the calculated results for Cu with experiment This denotes the advantage and efficiency of the present procedure for calculation and analysis of the EXAFS and its parameters including anharmonic contributions Acknowledgements One of the authors (N.V.H.) thanks the BUGH Wuppertal for financial support and hospitality The authors thank Professor J J Rehr for very helpful comments and for reading the manuscript of the paper before submission and Dr L Troăger for providing the data of high temperature EXAFS of Cu Useful discussions with Dr D LuătzenkirchenHecht are gratefully acknowledged Fig Comparison of the Fourier transform magnitude of EXAFS spectrum of Cu for the first shell for single scattering calculated by the present anharmonic theory to those of the harmonic model1) and of the experiment19) for T ¼ 297 K (a) and T ¼ 703 K (b) 1) J J Rehr, J Mustre de Leon, S I Zabinsky and R C Albers: J Am Chem Soc 113 (1991) 5135 2) B S Clausen, L Grab&k, H Tops(e, L B Hansen, P Stoltze, J K N(rskov and O H Nielsen: J Catal 141 (1993) 368 3) E D Crozier, J J Rehr and R Ingalls: X-ray absorption, ed D C Koningsberger and R Prins (Wiley, New York, 1988) Chap 4) J M Tranquada and R Ingalls: Phys Rev B 28 (1983) 3520 5) G Bunker: Nucl Instrum Methods 207 (1983) 437 6) L Wenzel, D Arvanitis, H Rabus, T Lederer and L Baberschke: Phys Rev Lett 64 (1990) 1765 7) E A Stern, P Livins 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include anharmonic contributions at. .. Cu at 295 K and 700 K calculated by the harmonic FEFF code1) and those including anharmonic contributions The anharmonic spectra are shifted to the left and attenuated especially at high k-values